Question
The 3rd term of an arithmetic sequence is 32
. The common difference is 12
. Complete the statement.(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is
and the recursive formula for the sequence is
.
Question 2
5. The first time Mya competed in a 500 meter free style race, her time was 15 minutes flat. Each subsequent race, Mya's time was reduced by 10 seconds.
Complete the statement.
(2 points)
The context is best represented by
an arithmetic sequence
and the domain is
continuous
. The common difference is 12
. Complete the statement.(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is
and the recursive formula for the sequence is
.
Question 2
5. The first time Mya competed in a 500 meter free style race, her time was 15 minutes flat. Each subsequent race, Mya's time was reduced by 10 seconds.
Complete the statement.
(2 points)
The context is best represented by
an arithmetic sequence
and the domain is
continuous
Answers
Answered by
GPT-4o mini
For the arithmetic sequence with the 3rd term as 32 and a common difference of 12, we can find the explicit and recursive formulas.
**1. Explicit and Recursive Formulas:**
- **Explicit Formula:** The n-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1)d
\]
Given that the 3rd term \(a_3 = 32\) and the common difference \(d = 12\):
\[
32 = a_1 + (3 - 1) \cdot 12 \\
32 = a_1 + 24 \\
a_1 = 32 - 24 \\
a_1 = 8
\]
Therefore, the explicit formula is:
\[
a_n = 8 + (n - 1) \cdot 12
\]
- **Recursive Formula:** The recursive formula for an arithmetic sequence has the form:
\[
a_n = a_{n-1} + d
\]
Starting with the first term \(a_1 = 8\), the recursive formula is:
\[
a_1 = 8 \\
a_n = a_{n-1} + 12 \quad (n > 1)
\]
So, the complete statement is:
- The explicit formula for the sequence is **a_n = 8 + (n - 1)12** and the recursive formula for the sequence is **a_1 = 8; a_n = a_{n-1} + 12 (n > 1)**.
---
**2. Context of Mya's Race:**
Given that Mya's first race time was 15 minutes (or 900 seconds) and she improves her time by 10 seconds for each subsequent race, we can represent the situation as follows:
- **Context Representation:** This scenario can be described as **an arithmetic sequence** since Mya's race times decrease by a constant value (10 seconds) each time.
- **Domain Representation:** Since Mya's race times can be recorded as discrete intervals (each race), but the times themselves are measurable in seconds and can vary continuously, we can say the domain is **continuous**.
So, the complete statement is:
- The context is best represented by **an arithmetic sequence** and the domain is **continuous**.
**1. Explicit and Recursive Formulas:**
- **Explicit Formula:** The n-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1)d
\]
Given that the 3rd term \(a_3 = 32\) and the common difference \(d = 12\):
\[
32 = a_1 + (3 - 1) \cdot 12 \\
32 = a_1 + 24 \\
a_1 = 32 - 24 \\
a_1 = 8
\]
Therefore, the explicit formula is:
\[
a_n = 8 + (n - 1) \cdot 12
\]
- **Recursive Formula:** The recursive formula for an arithmetic sequence has the form:
\[
a_n = a_{n-1} + d
\]
Starting with the first term \(a_1 = 8\), the recursive formula is:
\[
a_1 = 8 \\
a_n = a_{n-1} + 12 \quad (n > 1)
\]
So, the complete statement is:
- The explicit formula for the sequence is **a_n = 8 + (n - 1)12** and the recursive formula for the sequence is **a_1 = 8; a_n = a_{n-1} + 12 (n > 1)**.
---
**2. Context of Mya's Race:**
Given that Mya's first race time was 15 minutes (or 900 seconds) and she improves her time by 10 seconds for each subsequent race, we can represent the situation as follows:
- **Context Representation:** This scenario can be described as **an arithmetic sequence** since Mya's race times decrease by a constant value (10 seconds) each time.
- **Domain Representation:** Since Mya's race times can be recorded as discrete intervals (each race), but the times themselves are measurable in seconds and can vary continuously, we can say the domain is **continuous**.
So, the complete statement is:
- The context is best represented by **an arithmetic sequence** and the domain is **continuous**.